# A Generalization of Resource-Bounded Measure, With an Application (Extended Abstract)

@inproceedings{Buhrman1998AGO, title={A Generalization of Resource-Bounded Measure, With an Application (Extended Abstract)}, author={Harry Buhrman and Dieter van Melkebeek and Kenneth W. Regan and D. Sivakumar and Martin Strauss}, booktitle={STACS}, year={1998} }

We introduce resource-bounded betting games, and propose a generalization of Lutz's resource-bounded measure in which the choice of next string to bet on is fully adaptive. Lutz's martingales are equivalent to betting games constrained to bet on strings in lexicographic order. We show that if strong pseudo-random number generators exist, then betting games are equivalent to martingales, for measure on E and EXP. However, we construct betting games that succeed on certain classes whose Lutz…

## 14 Citations

### On the Autoreducibility of Random Sequences

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- 2003

The somewhat counterintuitive result is obtained that every algorithmically random language is polynomial-time i.o. autoreducible where the autoreducing machine poses its queries in a "quasi-nonadaptive" way.

### Bias Invariance of Small Upper Spans1

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- 2007

The resource-bounded measures of certain classes of languages are shown to be invariant under certain changes in the underlying probability measure. Speciically, for any real number > 0, any…

### Probabilistic martingales and BPTIME classes

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This paper gives a sufficient condition in terms of simulation by "decisive" probabilistic martingales that implies not only measure conservation, but also a much tighter bounded error Probabilistic time hierarchy than is currently known.

### Bias Invariance of Small Upper Spans

- MathematicsSTACS
- 2000

The resource-bounded measures of certain classes of languages are shown to be invariant under certain changes in the underlying probability measure, and a new, improved positive bias reduction of one bias sequence to another is introduced.

### Resource-Bounded Measure and Learnability

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- 2000

A nonuniformly computable variant of resource-bounded measure is introduced and it is shown that, for every fixed polynomial q, anyPolynomial-time learnable subclass of circuits of size q has measure zero with respect to P/poly.

### A generalization of Lutz's measure to probabilistic classes

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2002

A conditional time hierarchy theorem for probabilistic classes is proved, and it is shown that under the same assumption, both the class of p T-autoreducible sets for EXP have measure zero in BPE.

### Randomness vs. Completeness: On the Diagonalization Strength of Resource-Bounded Random Sets

- MathematicsMFCS
- 1998

We show that the question of whether the p-tt-complete or p-T-complete sets for the deterministic time classes E and EXP have measure 0 in these classes in the sense of Lutz's resource-bounded…

### Twelve Problems in Resource-Bounded Measure

- GeologyBull. EATCS
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A more recent snapshot of resource-bounded measure is given, focusing not so much on what has been achieved to date as on what the authors hope will be achieved in the near future.

### Hard Sets Are Hard to Find

- Mathematics
- 2006

We investigate the frequency of complete sets for t•ar·ious complexity classes within EXP under several polynomial-time reductions in the sense of resource bounded measure. We show that these sets…

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